MATH1220: Midterm 1 Study Guide
The following is an overview of the material that will be covered on the first exam.
§6.1 The Natural Logarithm Function
• The definition of the natural logarithm, including its derivative.
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• Computing integrals of the form du/u.
• Using properties of logarithms to simplify the computation of derivatives (this is called
logarithmic differentiation).
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• Computing certain trig integrals (e.g., tan x dx).
§6.2 Inverse Functions and Their Derivatives
• Finding the inverse of a function.
• Show a function has an inverse (without actually finding it). Our standard method for
doing this is using Theorem A from §6.2.
• Checking that two functions are inverses of each other (we just show that f ◦ f −1 (y) = y
and f −1 ◦ f (x) = x).
• Using the Inverse Function Theorem.
§6.3 The Natural Exponential Function
• The definition of the natural exponential function, including its derivative.
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• Computing derivatives of the form Dx (eu ) and integrals of the form eu du.
§6.4 General Exponential and Logarithmic Functions
• Derivatives and integrals involving general exponential functions (i.e., ax for arbitrary
a) and general logarithms (loga x).
• Differentiating (or integrating) using the definition of ax (e.g.,Dx (xx ) = Dx (ex ln x )).
§6.5 Exponential Growth and Decay
• Solving word problems involving exponential growth/decay.
• Know that limh→0 (1 + h)1/h = e.
• Solving separable differential equations by integration.
§6.6 First Order Linear Differential Equations
• Solving linear first-order differential equations using the integrating factor technique (I
guarantee you will be asked to do this on the exam).
• Finding the general solution to such a differential equation.
• Finding a specific solution using given initial conditions.
§6.7 Approximations for Differential Equations
• Sketch a specific solution to a differential equation when given the slope field and an
initial condition.
• Use Euler’s Method to approximate a solution to a differential equation.
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